Extrasolar Planets: Homework 3

This homework must be submitted on paper or in PDF format via E-mail (mwrsps@rit.edu) before 17:00 on Wedneday, Nov 13.

This homework is based on the recent paper JWST/NIRISS reveals the water-rich "steam world" atmosphere of GJ 9827 d by Piaulet-Ghorayeb et al., ApJ 974, L10, 2024. Please have a copy of that paper handy, and scan it over before you attempt to answer these questions.

    1. What is the radius of the host star, GJ 9827? Please express in meters.

      2. R(s) = 4.04 x 108 m

    2. What is the radius of the planet, GJ 9827d? Please express in meters.

      3. R(p) = 1.26 x 107 m

    3. Based on these radii, how deep should the transits be? Your result should be a small number, less than 1.

      4. depth = 0.00098

    4. Look at Figure 2 of the paper; in particular, the "white light" panels. How deep are the transits?

      5. obs depth ~ 0.001

    5. Does the observation match your calculation?

      6. yes!

  2. JWST observed this system during two transits. What was the total exposure time on the system while it was inside a transit? Don't include the time before or after transits. Express this total "in-transit exposure time" in seconds. 

    
         in-transit time =  1.3 hours + 1.3 hours = 2.6 hours 
     
                         =  9,360 seconds

    1. How many independent pipelines did the authors use to reduce the JWST measurements? Name each one.

      2. 2 pipelines: supreme-SPOON and NAMELESS

    2. How many independent codes did the authors use to make models of the atmospheres and their spectra? Name each one.

      3. 2 codes: SCARLET and POSEIDON

    3. Why didn't the authors use just one pipeline, or just one atmospheric code? Try to explain simply.

      4. By using two independent methods, they can verify that the results are robust, or discover some limitation of the code (or the data!) if the results don't match.

    1. Please look at Figure 3 to answer this question. How broad are the spectroscopic features of H2O? An approximate answer is okay. Express your answer in Angstroms.

      2. width is about 0.1 - 0.2 microns = 1000 - 2000 Angstroms

    2. Please look at Figure 2 to answer this question. How broad are the "spectroscopic bins" used to make light curves? Look at the wavelengths listed in the lower panels of this figure. Express your answer in Angstroms.

      3. bins are about 0.02 microns wide = 200 Angstroms

    3. In order to resolve a spectral feature well, one should sample it with at least 2 or 3 bins across its width. Can the authors use their measurements to resolve the water vapor features in the spectra?

      4. yes, the bins are at most 1/4 the width of the lines

  5. In Figure 3, how strong are the features of water vapor? That is, what is the amplitude of vertical variation in the spectrum due to the lines? Express your answer as a fraction of the continuum intensity. 
    
             features about 30 ppm  =  0.000 03  =  3 x 10^(-5)

  6. Let's do a simplified version of a signal-to-noise calculation for these observations. If one collects N photons in one bin during an exposure, and if the only source of noise is Poisson statistics in the incoming photons, then the uncertainty in the number of photons is √(N).

    In that case, the fractional uncertainty is 

    
                                        sqrt(N)       1
            fractional uncertainty  =   -------  =  -------
                                           N        sqrt(N)

    So, for example, if one collects N = 100 photons, one expects random fluctuations in the number to be about +/- 10 photons, which means the fractional uncertainty will be about 0.1, or one-tenth.

    Your answer to question 5 was a small quantity, much less than 1. Assume that you are trying to measure a feature in the light curve which is this small. How many photons N must you collect so that the fractional uncertainty in your counts is as small as the amplitude of these water vapor features?

    (Hint: the number N should be very large -- much larger than 100 or 1,000.) 

    

                 [         1        ]^2                  
           N  =  [ ---------------  ]    =~  1 x 10^(9)
                 [   3 x 10^(-5)    ]

  7. The star GJ 9827, observed in the near-infrared around 2 microns, produces a flux of roughly f = 11.5 photons per sq.cm. per second in bins of the size you calculated in question 4b. Suppose you observe with JWST, and assume the telescope and all instruments are 100% efficient.
    1. How many photons will you collect per second in each bin? 

      2. 
  
           Assume JWST circle with diameter 6.5 meters, area A = 33.2 sq.m.
 
                  A  =  33.2 sq.m. x 10^(4) (sq.cm/sq.m) =  3.32 x 10^5 sq.cm.

                  N  =  11.5 photon/s-sq.cm *  3.32 x 10^5 sq.cm.  

                     =  3.82 x 10^(6) photons/sec
    2. In order to collect N photons (the answer to question 6), for how many seconds would you have to collect photons with JWST? 

      3. 
                             1 x 10^9  photons
              time T  =  ---------------------------   =   262 seconds
                           3.82 x 10^6  photons/sec
    3. What was the actual total exposure time during transits that JWST observed this system? Look at your answer to question 2.

      4. actual exposure time = 9,360 seconds

    4. Did the JWST observations provide enough photons, and so enough signal-to-noise, to detect the very weak features in the spectrum that are shown in this paper? (I hope that you find the answer is "yes")

      5. yes!

If you have questions about this homework, or you aren't sure what to do, please contact me so I can help.